Problem: Let
\[x^5 - x^2 - x - 1 = p_1(x) p_2(x) \dotsm p_k(x),\]where each non-constant polynomial $p_i(x)$ is monic with integer coefficients, and cannot be factored further over the integers.  Compute $p_1(2) + p_2(2) + \dots + p_k(2).$
Solution: We can factor by pairing $x^5$ and $-x,$ and $-x^2$ and $-1$:
\begin{align*}
x^5 - x^2 - x - 1 &= (x^5 - x) - (x^2 + 1) \\
&= x(x^4 - 1) - (x^2 + 1) \\
&= x(x^2 + 1)(x^2 - 1) - (x^2 + 1) \\
&= (x^2 + 1)(x^3 - x - 1).
\end{align*}If $x^3 - x - 1$ factors further, then it must have a linear factor, which means it has an integer root.  By the Integer Root Theorem, the only possible integer roots are $\pm 1,$ and neither of these work, so $x^3 - x - 1$ is irreducible.

Thus, $(x^2 + 1)(x^3 - x - 1)$ is the complete factorization.  Evaluating each factor at 2, we get $(2^2 + 1) + (2^3 - 2 - 1) = \boxed{10}.$